1 - Introduction

Summary

The goal of this book is to exhibit the profound and myriad interrelations between the mathematics of modular forms and the physics of string theory. Our presentation is intended to be informal but mathematically precise, logically complete, and reasonably self-contained. The exposition is kept as simple as possible so as to be accessible to adventurous undergraduates, motivated graduate students, and dedicated professionals interested in the interface between theoretical physics and pure mathematics. Assuming little more than a knowledge of complex function theory, we introduce elliptic functions and elliptic curves as a lead-in to modular forms and their various deep generalizations. Following an economical introduction to string theory, its perturbative expansion, toroidal compactification, and supergravity limit are used to illustrate the power of modular invariance in physics. Dualities and their realization via modular forms in Yang–Mills theories with extended supersymmetry are studied both via the Seiberg–Witten solution and via their superconformal phase. Appendices are included to review foundational topics, and 75 exercises with detailed solutions give the reader ample opportunity for practice.